# A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint

## Abstract

**:**

## 1. Introduction

## 2. Notation and Fundamental Properties

**Lemma**

**1.**

**Proof.**

## 3. Lagrangian Formulation of Fourth-Order Dynamical Systems on Manifold

**Theorem**

**1.**

**Proof.**

## 4. Non-Conservative Forcing Terms and Reduced-Order Systems

## 5. An Invariant of Motion for Fourth-Order Conservative Systems

**Theorem**

**2.**

**Proof.**

## 6. Numerical Recipe to Simulate a Fourth-Order Dynamical System on Manifold

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Fiori, S.
A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint. *Mathematics* **2024**, *12*, 428.
https://doi.org/10.3390/math12030428

**AMA Style**

Fiori S.
A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint. *Mathematics*. 2024; 12(3):428.
https://doi.org/10.3390/math12030428

**Chicago/Turabian Style**

Fiori, Simone.
2024. "A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint" *Mathematics* 12, no. 3: 428.
https://doi.org/10.3390/math12030428